Subgroups — C₁₃₁×#29812 #45715696

Order 33536.

824 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#433×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2621×C₁₃₁	C₂	1	2	yes	maximal
4	#2056×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#2001×C₁₃₁	C₂	1	4	yes	maximal
6	#1960×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1835×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₁₃₁	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	4	yes	—
11	C₂×#117×C₁₃₁	C₂²	1	1	yes	characteristic
12	#494×C₁₃₁	—	2	4	no	—
13	#433×C₁₃₁	—	2	4	no	—
14	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂×#103×C₁₃₁	C₂²	1	8	yes	—
16	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#398×C₁₃₁	—	2	4	no	—
20	#397×C₁₃₁	—	2	4	no	—
21	#391×C₁₃₁	C₂²	1	4	yes	—
22	#382×C₁₃₁	C₂²	1	2	yes	—
23	#340×C₁₃₁	C₂²	1	4	yes	—
24	C₂²×Q₈×C₁₃₁	C₂³	1	4	yes	—
25	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
26	C₂×Q₁₆×C₁₃₁	—	2	8	no	—
27	C₂×Q₁₆×C₁₃₁	—	4	8	no	—
28	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
31	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
32	C₂²⋊Q₈×C₁₃₁	—	4	8	no	—
33	#117×C₁₃₁	—	2	2	no	—
34	#117×C₁₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₁₃₁	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₁₃₁	—	2	4	no	—
44	#103×C₁₃₁	—	4	8	no	—
45	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
46	#97×C₁₃₁	—	2	2	no	—
47	C₄×C₈×C₁₃₁	D₄	1	4	yes	—
48	#95×C₁₃₁	—	2	8	no	—
49	#95×C₁₃₁	—	2	4	no	—
50	C₂⁴×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₃₁	—	2	4	no	—
52	C₂×Q₈×C₁₃₁	—	2	4	no	—
53	C₂×Q₈×C₁₃₁	—	2	8	no	—
54	C₂×Q₈×C₁₃₁	—	2	8	no	—
55	C₂²×C₄×C₁₃₁	—	2	1	no	—
56	C₂²×C₄×C₁₃₁	—	2	1	no	—
57	C₂²×C₄×C₁₃₁	—	2	4	no	—
58	C₂²×C₄×C₁₃₁	—	2	4	no	—
59	C₂²×C₄×C₁₃₁	—	2	4	no	—
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
64	Q₁₆×C₁₃₁	—	4	16	no	—
65	C₂×C₈×C₁₃₁	—	2	4	no	—
66	C₂×C₈×C₁₃₁	—	2	2	no	—
67	C₂×C₈×C₁₃₁	—	2	4	no	—
68	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
69	C₄⋊C₄×C₁₃₁	—	2	4	no	—
70	C₄⋊C₄×C₁₃₁	—	2	8	no	—
71	C₄⋊C₄×C₁₃₁	—	4	8	no	—
72	C₄⋊C₄×C₁₃₁	—	4	8	no	—
73	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
74	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
75	C₄²×C₁₃₁	—	2	1	no	—
76	C₄²×C₁₃₁	—	2	1	no	—
77	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃₁	#41	1	2	yes	—
79	C₂³×C₁₃₁	—	2	1	no	—
80	C₂³×C₁₃₁	—	2	2	no	—
81	C₂³×C₁₃₁	—	2	4	no	—
82	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₁₃₁	—	4	8	no	—
84	Q₈×C₁₃₁	—	4	8	no	—
85	C₂×C₄×C₁₃₁	—	2	4	no	—
86	C₂×C₄×C₁₃₁	—	2	2	no	—
87	C₂×C₄×C₁₃₁	—	2	1	no	—
88	C₂×C₄×C₁₃₁	—	2	2	no	—
89	C₂×C₄×C₁₃₁	—	2	4	no	—
90	C₂×C₄×C₁₃₁	—	4	2	no	—
91	C₂×C₄×C₁₃₁	—	4	2	no	—
92	C₂×C₄×C₁₃₁	—	4	4	no	—
93	C₂×C₄×C₁₃₁	—	4	4	no	—
94	C₂×C₄×C₁₃₁	—	4	4	no	—
95	C₂×C₄×C₁₃₁	—	4	4	no	—
96	C₂×C₄×C₁₃₁	—	4	8	no	—
97	C₂×C₄×C₁₃₁	#121	1	4	yes	—
98	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
101	C₈×C₁₃₁	—	2	4	no	—
102	C₈×C₁₃₁	—	4	2	no	—
103	C₂²×C₁₃₁	—	4	1	no	—
104	C₂²×C₁₃₁	—	4	4	no	—
105	C₂²×C₁₃₁	—	4	2	no	—
106	C₂²×C₁₃₁	C₂×#121	1	1	yes	characteristic
107	C₂²×C₁₃₁	#535	1	2	yes	—
108	C₂²×C₁₃₁	#578	1	4	yes	—
109	C₄×C₁₃₁	—	2	4	no	—
110	C₄×C₁₃₁	—	4	1	no	—
111	C₄×C₁₃₁	—	4	2	no	—
112	C₄×C₁₃₁	—	8	4	no	—
113	C₄×C₁₃₁	—	8	4	no	—
114	C₄×C₁₃₁	#535	1	2	yes	—
115	C₄×C₁₃₁	#540	1	2	yes	—
116	C₂×C₁₃₁	—	8	1	no	—
117	C₂×C₁₃₁	#2619	1	1	yes	characteristic
118	C₂×C₁₃₁	#3059	1	2	yes	—
119	C₂×C₁₃₁	#3223	1	4	yes	—
120	#29812	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₃₁	#29812	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×#494	C₂×C₁₃₁	1	1	yes	characteristic, complemented
123	C₂×#433	C₂×C₁₃₁	1	1	yes	characteristic
124	#2621	C₂×C₁₃₁	1	2	yes	—
125	#2056	C₂×C₁₃₁	1	4	yes	complemented
126	#2001	C₂×C₁₃₁	1	4	yes	—
127	#1960	C₂×C₁₃₁	1	2	yes	complemented
128	#1835	C₂×C₁₃₁	1	1	yes	characteristic, complemented
129	C₂²×Q₁₆	C₂²×C₁₃₁	1	4	yes	—
130	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
131	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	4	yes	—
132	C₂×#117	C₂²×C₁₃₁	1	1	yes	characteristic
133	#494	—	2	4	no	—
134	#433	—	2	4	no	—
135	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
136	C₂×#103	C₂²×C₁₃₁	1	8	yes	—
137	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
138	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
140	#398	—	2	4	no	—
141	#397	—	2	4	no	—
142	#391	C₂²×C₁₃₁	1	4	yes	—
143	#382	C₂²×C₁₃₁	1	2	yes	—
144	#340	C₂²×C₁₃₁	1	4	yes	—
145	C₂²×Q₈	C₂³×C₁₃₁	1	4	yes	—
146	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	4	8	no	—
149	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
150	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
151	C₄⋊Q₈	—	2	4	no	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₂²⋊Q₈	—	4	8	no	—
154	#117	—	2	2	no	—
155	#117	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
160	C₂×C₂²⋊C₄	—	2	4	no	—
161	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	2	yes	—
162	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	#103	—	4	8	no	—
166	C₂²⋊C₈	—	2	4	no	—
167	#97	—	2	2	no	—
168	C₄×C₈	D₄×C₁₃₁	1	4	yes	—
169	#95	—	2	4	no	—
170	#95	—	2	8	no	—
171	C₂⁴	C₂×D₄×C₁₃₁	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
184	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
185	Q₁₆	—	4	16	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	—	4	8	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	4	8	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
199	C₄²	#41×C₁₃₁	1	2	yes	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	4	no	—
203	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
204	Q₈	—	4	8	no	—
205	Q₈	—	4	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	#121×C₁₃₁	1	4	yes	—
219	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
221	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
222	C₈	—	2	4	no	—
223	C₈	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	4	no	—
227	C₂²	C₂×#121×C₁₃₁	1	1	yes	characteristic
228	C₂²	#535×C₁₃₁	1	2	yes	—
229	C₂²	#578×C₁₃₁	1	4	yes	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	8	4	no	—
234	C₄	—	8	4	no	—
235	C₄	#535×C₁₃₁	1	2	yes	—
236	C₄	#540×C₁₃₁	1	2	yes	—
237	C₂	—	8	1	no	—
238	C₂	#2619×C₁₃₁	1	1	yes	characteristic
239	C₂	#3059×C₁₃₁	1	2	yes	—
240	C₂	#3223×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂⁴×C₁₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×#117×C₁₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#494×C₁₃₁	C₂	semidirect
4. #2056×C₁₃₁	C₂	semidirect
6. #1960×C₁₃₁	C₂	semidirect
7. #1835×C₁₃₁	C₂	semidirect
120. #29812	C₁₃₁	direct (N × H)
121. C₁₃₁	#29812	direct (N × H)
122. C₂×#494	C₂×C₁₃₁	semidirect
125. #2056	C₂×C₁₃₁	semidirect
127. #1960	C₂×C₁₃₁	semidirect
128. #1835	C₂×C₁₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#494×C₁₃₁	2
C₂×#433×C₁₃₁	2
#2621×C₁₃₁	2
#2056×C₁₃₁	2
#2001×C₁₃₁	2
#1960×C₁₃₁	2
#1835×C₁₃₁	2
#29812	131

Subgroups — C131×#29812 #45715696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#29812 #45715696